A103377 a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 128, 129, 136, 157, 192, 227

Offset

1,11

Comments

k=9 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, k=5 case is A103373, k=6 case is A103374, k=7 case is A103375, k=8 case is A103376, k=10 case is A103378 and k=11 case is A103379. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=9 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^10 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0757660660868371580595995241652758206925302476392. Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/10))^(1/10)))^(1/10))))^(1/10)))))^(1/10))))). The sequence of prime values in this k=9 case is A103387; The sequence of semiprime values in this k=9 case is A103397.

In analogy to the Fibonacci sequence, one might prefer to start this sequence with offset 0. - M. F. Hasler, Sep 19 2015

References

A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (1999) 229-245.

Links

Table of n, a(n) for n=1..78.

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

Richard Padovan, Dom Hans Van Der Laan And The Plastic Number.

E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 291-306

J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61(1988)1-16.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1,1).

Formula

a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 1 and for n>10: a(n) = a(n-9) + a(n-10).

O.g.f.: -x*(x^2+x+1)*(x^6+x^3+1)/(-1+x^9+x^10). - R. J. Mathar, May 02 2008

Example

a(83) = 257 because a(83) = a(83-9) + a(83-10). a(74) + a(73) = 129 + 128. This sequence has as elements 5, 17 and 257, which are all Fermat Primes.

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* Charles R Greathouse IV, Jan 11 2013 *)

Prog

(PARI) Vec((1+x+x^2)*(1+x^3+x^6)/(1-x^9-x^10)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

Crossrefs

Cf. A000045, A000931, A079398, A103372-A103376, A103378-A103380, A103387, A103397.

Sequence in context: A300358 A102682 A116371 * A343336 A103817 A073808

Adjacent sequences: A103374 A103375 A103376 * A103378 A103379 A103380

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 15 2005

Extensions

Edited by R. J. Mathar, May 02 2008

Edited by M. F. Hasler, Sep 19 2015

Status

approved